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A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
In computer programming, a macro (short for "macro instruction"; from Greek μακρο- 'long, large' [1]) is a rule or pattern that specifies how a certain input should be mapped to a replacement output. Applying a macro to an input is known as macro expansion. The input and output may be a sequence of lexical tokens or characters, or a syntax ...
A concatenative programming language is a point-free computer programming language in which all expressions denote functions, and the juxtaposition of expressions denotes function composition. [4] Concatenative programming replaces function application , which is common in other programming styles, with function composition as the default way ...
and so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disk centered at b if and only if its Taylor series converges to the value of the function at each point of the disk. If f (x) is equal to the sum of its Taylor series for all x in the complex plane, it is called entire.
In computing, inline expansion, or inlining, is a manual or compiler optimization that replaces a function call site with the body of the called function. Inline expansion is similar to macro expansion, but occurs during compilation, without changing the source code (the text), while macro expansion occurs prior to compilation, and results in different text that is then processed by the compiler.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
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In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point.